C*-algebras associated with Hilbert C*-quad modules of finite type

A Hilbert $C^*$-quad module of finite type has a multi structure of Hilbert $C^*$-bimodules with two finite bases. We will construct a $C^*$-algebra from a Hilbert $C^*$-quad module of finite type and prove its universality subject to certain relations among generators. Some examples of the $C^*$-algebras from Hilbert $C^*$-quad modules of finite type will be presented.


Introduction
G. Robertson-T. Steger [23] have initiated a certain study of higher dimensional analogue of Cuntz-Krieger algebras from the view point of tiling systems of 2dimensional plane. After their work, A. Kumjian-D. Pask [11] have generalized their construction to introduce the notion of higher rank graphs and its C * -algebras. Since then, there have been many studies on these C * -algebras by many authors (see for example [6], [7], [11], [22], [18], [23], etc.).
In [12], the author has introduced a notion of C * -symbolic dynamical system, which is a generalization of a finite labeled graph, a λ-graph system and an automorphism of a unital C * -algebra. It is denoted by (A, ρ, Σ) and consists of a finite family {ρ α } α∈Σ of endomorphisms of a unital C * -algebra A such that ρ α (Z A ) ⊂ Z A , α ∈ Σ and α∈Σ ρ α (1) ≥ 1 where Z A denotes the center of A. It provides a subshift Λ ρ over Σ and a Hilbert C * -bimodule H ρ A over A which gives rise to a C * -algebra O ρ as a Cuntz-Pimsner algebra ( [12], cf. [8], [16], [21]). In [13] and [14], the author has extended the notion of C * -symbolic dynamical system to C * -textile dynamical system which is a higher dimensional analogue of C * -symbolic dynamical system. A C * -textile dynamical system (A, ρ, η, Σ ρ , Σ η , κ) consists of two C * -symbolic dynamical systems (A, ρ, Σ ρ ) and (A, η, Σ η ) with common unital C * -algebra A and commutation relations κ between the endomorphisms ρ α , α ∈ Σ ρ and η a , a ∈ Σ η . A C * -textile dynamical system provides a two-dimensional subshift and a multi structure of Hilbert C * -bimodules that has multi right actions and multi left actions and multi inner products. Such a multi structure of Hilbert C * -bimodule is called a Hilbert C * -quad module. In [14], the author has introduced a C * -algebra associated with the Hilbert C * -quad module of C * -textile dynamical system. It is generated by the quotient images of creation operators on two-dimensional analogue of Fock Hilbert module by module maps of compact operators. As a result, the C * -algebra has been proved to have a universal property subject to certain operator relations of generators encoded by structure of C * -textile dynamical system ( [14]).
In this paper, we will generalize the construction of the C * -algebras of Hilbert C * -quad modules of C * -textile dynamical systems. Let A, B 1 , B 2 be unital C *algebras. Assume that A has unital embeddings into both B 1 and B 2 . A Hilbert C * -quad module H over (A; B 1 , B 2 ) is a Hilbert C * -bimodule over A with A-valued right inner product · | · A which has a multi structure of Hilbert C * -bimodules over B i with right actions ϕ i of B i and left actions φ i of B i and B i -valued inner products · | · Bi for i = 1, 2 satisfying certain compatibility conditions. A Hilbert C * -quad module H is said to be of finite type if there exist a finite basis {u 1 , . . . , u M } of H as a Hilbert C * -right module over B 1 and a finite basis {v 1 , . . . , v N } of H as a Hilbert C * -right module over B 2 such that for ξ, η ∈ H (see [26] for the original definition of finite basis of Hilbert module).
For a Hilbert C * -quad module, we will construct a Fock space F (H) from H, which is a 2-dimensional analogue to the ordinary Fock space of Hilbert C * -bimodules (cf. [10], [21]). We will then define two kinds of creation operators s ξ , t ξ for ξ ∈ H on F (H). The C * -algebra on F (H) generated by them is denoted by T F (H) and called the Toeplitz quad module algebra. We then define the C * -algebra O F (H) associated with the Hilbert C * -quad module H by the quotient C * -algebra of T F (H) by the ideal generated by the finite rank operators. We will then prove that the C *algebra O F (H) for a C * -quad module H of finite type has a universal property in the following way.
The eight relations of the operators above are called the relations (H). As a corollary we have The paper is organized in the following way. In Section 2, we will define Hilbert C * -quad module and present some basic properties. In Section 3, we will define a C * -algebra O F (H) from Hilbert C * -quad module H of general type by using creation operators on Fock Hilbert C * -quad module. In Section 4, we will study algebraic structure of the C * -algebra O F (H) for a Hilbert C * -quad module H of finite type. In Section 5, we will prove, as a main result of the paper, that the C * -algebra O F (H) has the universal property stated as in Theorem 1.1. A sterategy to prove Theorem 1.1 is to show that the C * -algebra O F (H) is regarded as a Cuntz-Pimsner algebra for a Hilbert C * -bimodule over the C * -algebra generated by φ 1 (B 1 ) and φ 2 (B 2 ). We will then prove the gauge invariant universarity of the C * -algebra (Theorem 5.16). In Section 6, we will present K-theory formulae for the C * -algebra O H . In Section 7, we will give examples. In Section 8, we will formulate higher dimensional anlogue of our situations and state a generalized proposition of Theorem 1.1 without proof.
Throughout the paper, we will denote by Z + the set of nonnegative integers and by N the set of positive integers.

Hilbert C * -quad modules
Throughout the paper we fix three unital C * -algebras A, B 1 , B 2 such that A ⊂ B 1 , A ⊂ B 2 with common units. We assume that there exists a right action Suppose that H is a Hilbert C * -bimodule over A, which has a right action of A, an A-valued right inner product · | · A and a * -homomorphism φ A from A to the algebra of all bounded adjointable right A-module maps L A (H) satisfying (i) · | · A is linear in the second variable.
if H has a further structure of a Hilbert C * -bimodule over B i for each i = 1, 2 with right action ϕ i of B i and left action φ i of B i and B i -valued right inner product · | · Bi such that for z ∈ B 1 , w ∈ B 2 for ξ ∈ H, z ∈ B 1 , w ∈ B 2 , a ∈ A and where A is regarded as a subalgebra of B i . The left action φ i of B i on H means that φ i (b i ) for b i ∈ B i is a bounded adjointable operator with respect to the inner product · | · Bi for each i = 1, 2. The operator φ i (b i ) for b i ∈ B i is also adjointable with respect to the inner product · | · A . We assume that the adjoint of φ i (b i ) with respect to the inner product · | · Bi coincides with the adjoint of φ i (b i ) with respect to the inner product · | · A . Both of them coincide with φ i (b * i ). We assume that the left actions φ i of B i on H for i = 1, 2 are faithful. We require the following compatibility conditions between the right A-module structure of H and the right A-module structure of B i through ψ i : We further assume that H is a full Hilbert C * -bimodule with respect to the three inner products · | · A , · | · B1 , · | · B2 for each. This means that the C *algebras generated by elements induced by the right inner products respectively. By definition, H is complete under the above three norms for each. Definition.
(i) A Hilbert C * -quad module H over (A; B 1 , B 2 ) is said to be of general type if there exists a faithful completely positive map λ i : such that for all ξ, η ∈ H. Following We note that for a Hilbert C * -quad module of general type, the conditions (2.6) imply Hence the identity operators from the Banach spaces (H, · Bi ) to (H, · A ) are bounded linear maps. By the inverse mapping theorem, there exist constants C ′ i such that ξ Bi ≤ C ′ i ξ A for ξ ∈ H. Therefore the three norms · A , · Bi , i = 1, 2, induced by the three inner products · | · A , · | · Bi , i = 1, 2 on H are equivalent to each other.
They give rise to faithful completely positive maps λ i : B i −→ A, i = 1, 2. The equalities (2.10) (2.11) imply that for ξ, η ∈ H, i = 1, 2. (2.14) It then follows that Since H is full, the equalities (2.5) hold. Proof. For ξ ∈ H, by the equalities We similarly have We present some examples.
for b i ∈ B i , a ∈ A. We set B 1 = B 2 = A. We put H α,β = A and equip it with Hilbert C * -quad module structure over (A; A, A) in the following way. For ξ = x, ξ ′ = x ′ ∈ H α,β = A, a ∈ A, z ∈ B 1 = A, w ∈ B 2 = A, define the right A-module structure and the right A-valued inner product · | · A by Define the right actions ϕ i of B i with right B i -valued inner products · | · Bi and the left actions φ i of B i by setting It is straightforward to see that H α,β is a Hilbert C * -quad module over (A; A, A) of strongly finite type.

2.
We fix natural numbers 1 < N, M ∈ N. Consider finite dimensional commutative C * -algebras A = C, B 1 = C N , B 2 = C M . The right actions ψ i of A on B i are naturally defined as right multiplications of C. The algebras B 1 , B 2 have the ordinary product structure and the inner product structure which we denote by · | · N and · | · M respectively. Let us denote by H M,N the tensor product C M ⊗ C N . Define the right actions ϕ i of B i with B i -valued right inner products · | · Bi and the left actions φ i of B i on H M,N = C M ⊗ C N for i = 1, 2 by setting . , M and f k , k = 1, . . . , N be the standard basis of C M and that of C N respectively. Put the finite bases It is straightforward to see that H M,N is a Hilbert C * -quad module over (C; C N , C M ) of strongly finite type.
3. Let (A, ρ, η, Σ ρ , Σ η , κ) be a C * -textile dynamical system which means that for j ∈ Σ η , l ∈ Σ ρ endomorphisms η j , ρ l of A are given with commutation relations [14] for its detail construction). The two triplets (A, ρ, Σ ρ ) and (A, η, Σ η ) are C * -symbolic dynamical systems ( [12]), that yield C * -algebras O ρ and O η respectively. The C * -algebras B 1 and B 2 are defined as the C * -subalgebra of O η generated by elements T j yT * j , j ∈ Σ η , y ∈ A and that of O ρ generated by S k yS * k , k ∈ Σ ρ , y ∈ A respectively. Define the maps which yield the right actions of A on B i , i = 1, 2. Define the maps λ i : We similarly have by putting We see that H ρ,η κ is a Hilbert C * -quad module of strongly finite type. In particular, two nonnegative commuting matrices A, B with a specification κ coming from the equality AB = BA yield a C * -textile dynamical system and hence a Hilbert C *quad module of strongly finite type, which are studied in [15].

Fock Hilbert C * -quad modules and creation operators
In this section, we will construct a C * -algebra from a Hilbert C * -quad module H of general type by using two kinds of creation operators on Fock space of Hilbert C * -quad module. We first consider relative tensor products of Hilbert C *quad modules and then introduce Fock space of Hilbert C * -quad modules which is a two-dimensional analogue of Fock space of Hilbert C * -bimodules. We fix a Hilbert C * -quad module H over (A; B 1 , B 2 ) of general type as in the preceding section. The Hilbert C * -quad module H is originally a Hilbert C * -right module over A with A-valued inner product · | · A . It has two other structure of Hilbert C * -bimodules, the Hilbert C * -bimodule (φ 1 , H, ϕ 1 ) over B 1 and the Hilbert This situation is written as in the figure: We will define two kinds of relative tensor products as Hilbert C * -quad modules over (A; B 1 , B 2 ). The latter one should be written vertically as H ⊗ B2 H rather than horizontally H ⊗ B2 H. The first relative tensor product H ⊗ B1 H is defined as the relative tensor product as Hilbert C * -modules over B 1 , where the left H is a right B 1 -module through ϕ 1 and the right H is a left B 1 -module through φ 1 . It has a right B 1 -valued inner product and a right B 2 -valued inner product defined by respectively. It has two right actions, id ⊗ ϕ 1 from B 1 and id ⊗ ϕ 2 from B 2 . It also has two left actions, φ 1 ⊗ id from B 1 and φ 2 ⊗ id from B 2 . By these operations H ⊗ B1 H is a Hilbert C * -bimodule over B 1 as well as a Hilbert C * -bimodule over B 2 . It also has a right A-valued inner product defined by We denote the above operations Similarly we consider the other relative tensor product H⊗ B2 H defined by the relative tensor product as Hilbert C * -modules over B 2 , where the left H is a right B 2 -module through ϕ 2 and the right H is a left B 2 -module through φ 2 . By a symmetric discussion to the above, H ⊗ B2 H is a Hilbert C * -quad module over (A; B 1 , B 2 ). The following lemma is routine.
yield isomorphisms of Hilbert C * -quad modules respectively.
We write the isomorphism class of the former Hilbert C * -quad modules as H 1 ⊗ B1 H 2 ⊗ B2 H 3 and that of the latter ones as Note that the direct sum B 1 ⊕B 2 has a structure of a pre Hilbert C * -right module over A by the following operations: By (2.5) the equality We denote the relative tensor product H ⊗ Bi H and elements ξ ⊗ Bi η by H ⊗ i H and ξ ⊗ i η respectively for i = 1, 2. Let us define the Fock Hilbert C * -quad module as a two-dimensional analogue of the Fock space of Hilbert C * -bimodules. Put Γ 0 = {∅} and Γ n = {(i 1 , . . . , i n )) | i j = 1, 2}, n = 1, 2, . . . . We set as Hilbert C * -bimodules over A. We will define the Fock Hilbert C * -module F (H) by setting For ξ ∈ H κ we define two operators by setting for n = 0, and for n = 1, 2, . . . , Lemma 3.2. For ξ ∈ H the two operators Proof. We will show the assertion for s ξ . For n = 0, we have for b 1 ⊕ b 2 ∈ B 1 ⊕ B 2 and a ∈ A, For n = 1, 2, . . . , we have It is clear that the two operators s ξ , t ξ yield bounded right A-module maps on F (H) having its adjoints with respect to the A-valued right inner product on F (H). The operators are still denoted by s ξ , t ξ respectively. The adjoints of s ξ , t ξ : Proof. We will show the assertions (i) and (ii) for Denote byφ i the left actions of B i , i = 1, 2 on F n (H) and hence on F (H) respectively. They satisfy the following equalities More generally let us denote by L A (H) and L A (F (H)) the C * -algebras of all bounded adjointable right A-module maps on H and on F (H) with respect to their right A-valued inner products respectively. For L ∈ L A (H), define L ∈ L A (F (H)) by Lemma 3.5. For ξ, ζ ∈ H, z ∈ B 1 , w ∈ B 2 , L ∈ L A (H) and c, d ∈ C, the following equalities hold on F (H): Proof. The equalities (3.1) are obvious. We will show the equalities (3.2) and (3.3) for s ξ . We have for so that s Lξϕ1(z) = Ls ξφ1 (z) on F n (H), n = 0, 1, . . . . Hence the equalities (3.2) hold. The C * -subalgebra of L A (F (H)) generated by the operators s ξ , t ξ for ξ ∈ H is denoted by T F (H) and is called the Toeplitz quad module algebra for H.
Lemma 3.7. There exists an action γ of R/Z = T on T F (H) such that Proof. We will first define a one-parameter unitary group u r , r ∈ R/Z = T on F (H) with respect to the right A-valued inner product as in the following way. For n = 0 : u r : For n = 1, 2, . . . : u r : F n (H) −→ F n (H) is defined by We therefore have a one-parameter unitary group u r on F (H). We then define an automorphism γ r on L A (F (H)) for r ∈ R/Z by
Therefore we conclude that γ r (s ξ ) = e 2π √ −1r s ξ on F (H) and similarly γ r (t ξ ) = e 2π √ −1r t ξ on F (H). It is direct to see that It is also obvious that γ r (T F (H) ) = T F (H) for r ∈ R/Z.
Denote by J(H) the C * -subalgebra of L A (F (H)) generated by the elements The algebra J(H) is a closed two-sided ideal of L A (F (H)).
Definition. The C * -algebra O F (H) associated to the Hilbert C * -quad module H of general type is defined by the quotient C * -algebra of T for ξ ∈ H and z ∈ B 1 , w ∈ B 2 . By the preceding lemmas, we have Proposition 3.8. The C * -algebra O F (H) is generated by the family of operators S ξ , T ξ for ξ ∈ H. It contains the operators Φ 1 (z), Φ 2 (w) for z ∈ B 1 , w ∈ B 2 . They satisfy the following equalities for ξ, ζ ∈ H, c, d ∈ C and z, z ′ ∈ B 1 , w, w ′ ∈ B 2 .

Define two projections on F (H) by
As Therefore we conclude that M i=1 s i s * i = P s + P 1 and similarly N k=1 t k t * k = P t + P 1 .
We set the operators T k T * k = 1.
(ii) The first equality of (4.6) is (4.5). As the projection P 0 belongs to J(H), Lemma 4.1 ensures us the second equality of (4.6). The equalities (4.7) come from (3.7). For z ∈ B 1 and j = 1, . . . , M, we have which goes to the first equality of (4.8). The other equalities of (4.8) and (4.9) are similarly shown.
(iii) The assertion is direct from Lemma 3.7.
The action γ of T on O F (H) defined in the above theorem (iii) is called the gauge action.

5.
The universal C * -algebras associated with Hilbert C * -quad modules In this section, we will prove that the C * -algebra O F (H) associated with a Hilbert C * -quad module of finite type is the universal C * -algebra subject to the operator relations stated in Theorem 4.3 (ii). Throughout this section, we fix a Hilbert C *quad module H over (A; B 1  Let P H be the universal * -algebra generated by operators S 1 , . . . , S M , T 1 , . . . , T N and elements z ∈ B 1 , w ∈ B 2 subject to the relations: By the relations (5.1), one sees that 0 ≤ P, Q ≤ 1 and P + Q = 1, P Q = 0. It is easy to see that both P and Q are projections.

Lemma 5.2.
(i) For i, j = 1, . . . , M and z ∈ B 1 , w ∈ B 2 we have For k, l = 1, . . . , N and z ∈ B 1 , w ∈ B 2 we have Proof. (i) By (5.3), we have The other equality S * i wS j = u i | φ 2 (w)u j B1 is similarly shown to the above equalities.
(ii) is similar to (i).
(i) For w ∈ B 2 , j = 1, . . . M , the element S * j wS j belongs to A and the formula holds: (ii) For z ∈ B 1 , l = 1, . . . N , the element T * l zT l belongs to A and the formula holds: Lemma 5.4. The following equalities for z ∈ B 1 and w ∈ B 2 hold: Proof. (i) By (5.3) and (5.4), we have Similarly we have (5.6).
(iii) By (i) we have As S * j T m = T * l S g = 0 for any j, g = 1, . . . , M, l, m = 1, . . . , N , it follows that Lemma 5.5. Let p(z, w) be a polynomial of elements of B 1 and B 2 . Then we have Proof. For z ∈ B 1 , w ∈ B 2 and i, j = 1, . . . , M , by putting so that the assertion of (i) holds. (ii) is similarly shown to (i). Proof. (i) By the previous lemma, we know As u i | u h B1 z h belongs to B 1 , we see the assertion. (ii) is similarly shown to (i).
(iii) As T * k S i = 0, we have (iv) is similarly shown to (i).
Proof. The assertion follows from the preceding lemmas.
By construction, every representation of B 1 and B 2 on a Hilbert space H together with operators S i , i = 1, . . . , M, T k , k = 1, . . . , N satisfying the relations (H) extends to a representation of P H on B(H). We will endow P H with the norm obtained by taking the supremum of the norms in B(H) over all such representations. Note that this supremum is finite for every element of P H because of the inequalities S i , T k ≤ 1, which come from (5.1). The completion of the algebra P H under the norm becomes a C * -algebra denoted by O H , which is called the universal C * -algebra subject to the relations (H). Denote by C * (φ 1 (B 1 ), φ 2 (B 2 )) the C * -subalgebra of L A (H) generated by φ 1 (B 1 ) and φ 2 (B 2 ).
Lemma 5.8. An element L of the C * -algebra C * (φ 1 (B 1 ), φ 2 (B 2 )) is both a right B 1 -module map and a right B 2 -module map. This means that the equalities Proof. Since both the operators φ 1 (z) for z ∈ B 1 and φ 2 (w) for w ∈ B 2 are right B i -module maps for i = 1, 2, any element of the * -algebra algebraically generated by φ 1 (B 1 ) and φ 2 (B 2 ) is both a right B 1 -module map and a right B 2 -module map.
Hence it is easy to see that any element L of the C * -algebra C * (φ 1 (B 1 ), φ 2 (B 2 )) is both a right B 1 -module map and a right B 2 -module map. 20 Denote by B • the C * -subalgebra of O H generated by B 1 and B 2 .
The following theorem is one of the main results of the paper.

K-Theory formulae
Let H be a Hilbert C * -quad module over (A; B 1 , B 2 ) of finite type as in the preceding section. In this section, we will state K-theory formulae for the C *algebra O F (H) . By the previous section, the C * -algebra O F (H) is regarded as the universal C * -algebra O H generated by the operators S 1 , . . . , S M and T 1 , . . . , T N and the elements z ∈ B 1 and w ∈ B 2 subject to the relations (H). Let us denote by B • the C * -subalgebra of O H generated by elements z ∈ B 1 and w ∈ B 2 . By Lemma 5.9 the correspondence z, w ∈ B • −→ φ 1 (z), φ 2 (w) ∈ C * (φ 1 (B 1 ), φ 2 (B 2 )) ⊂ L A (H) (6.1) gives rise to a * -isomorphism from B • onto C * (φ 1 (B 1 ), φ 2 (B 2 )) as C * -algebras, which is denoted by φ • . We will restrict our interest to the case when As in the argument of [17], O H × h T is stably isomorphic to F H . Hence we have K * (O H × h T) is isomorphic to K * (F H ). The dual action h induces an automorphism on the group K * (O H × h T) and hence on K * (F H ), which is denoted by σ * . Then by [17] (cf. [3], [21], etc.) we have Proposition 6.1. The following six term exact sequence of K-theory hold: We put for x ∈ B • λ 1,i (x) = S * i xS i , i = 1, . . . , M, λ 2,k (x) = T * k xT k , k = 1, . . . , N.
Proof. By the preceding lemma, one knows that e (i,k) , S i , T k are generated by the operators S (i,k) , T (i,k) so that the algebra O HM,N is generated by the partial isometries S (i,k) , T (i,k) , (i, k) ∈ Σ • .
Let I n be the n × n identity matrix and E n the n × n matrix whose entries are all 1 ′ s.